Polynomial interpolation in several variables: lattices, differences, and ideals

When passing from one to several variables, the nature and structure of polynomial interpolation changes completely: the solvability of the interpolation problem with respect to a given finite dimensional polynomial space, like all polynomials of at most a certain total degree, depends not only on the number, but significantly on the geometry of the nodes. Thus the construction of interpolation sites suitable for a given space of polynomials or of appropriate interpolation spaces for a given set of nodes become challenging and nontrivial problems. The paper will review some of the basic constructions of interpolation lattices which emerge from the geometric characterization due to Chung and Yao. Depending on the structure of the interpolation problem, there are different representations of the interpolation polynomial and several formulas for the error of interpolation, reflecting the underlying point geometry and employing different types of differences, divided and non–divided ones. In addition, we point out the close relationship with constructive ideal theory and degree reducing interpolation, whose most prominent representer is the least interpolant, introduced by de Boor et al.